Sato's Tau-functions expressed by Weyl-functions and its application to KdV flow (Tosio Kato Centennial Conference)
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(2) 56. to construct solutions to the. \mathrm{K}\mathrm{d}\mathrm{V}. equation, although it seems that his method. also has difficulty to go beyond the class investigated by [SW]. In [Ko2] we gave a representation of the Tau‐fUnctions by the Weyl functions of Schrödinger operators. Since the Weyl function is quantity defined for general potentials, there is a hope for this representation to give general solutions to the \mathrm{K}\mathrm{d}\mathrm{V} equation. In this paper we give a brief sketch of the proof of the construction of a \mathrm{K}\mathrm{d}\mathrm{V} flow on a space containing the Schwartz space S(R) and smooth almost periodic functions. To state the results we prepare several notions from spectral theory of Schrödinger operators. For a real valued q \in L^{1}(R) assume that the associ‐ ated Schrödinger operator L_{q}. L_{q}f\triangle=-\partial_{x}^{2}f+qf. (1). is essentially self‐adjoint, which is equivalent to the unique existence of non‐ trivial solutions. f\pm \mathrm{t}\mathrm{o}L_{q}f=zf with f\pm\in L^{2}(R_{\pm}). Its Weyl functions. m\pm. and. f\pm(0)=1. for z\in. C\backslash R.. are defined by. m\displaystyle \pm(z)=\pm\frac{f_{\pm}'(0,z)}{f\pm(0,z)}.. m\pm are holomorphic on C\backslash R and have positive imaginary parts (such a func‐ tion is called Herglotz function). the inverse spectral theory implies that m\pm uniquely recover a potential q . A potential q is called reflectionless on F\in \mathcal{B}(R). if its Weyl functions. m\pm. satisfy. m+( $\xi$+i0)=-\overline{m_{-}( $\xi$+i0)} \mathrm{a}.\mathrm{e}. $\xi$\in F .. (2). m(z)=\left\{ begin{ar ay}{l} -m+(-z^{2})&\mathrm{i}\mathrm{f}{\rmRe}z>0\ m_{-}(z^{2})&\mathrm{i}\mathrm{f}{\rmRe}z<0 \end{ar ay}\right.. (3). Set. and assume that there exist $\lambda$_{0}<0<$\lambda$_{1} such that. \mathrm{i}\mathrm{n}\mathrm{f}\mathrm{s}\mathrm{p}L_{q}>$\lambda$_{0} , and Then, m is holomorphic on C\backslash a simple pole at \infty such that. q. is reflectionless on ($\lambda$_{1}, \infty) .. ([-\sqrt{-$\lambda$_{0} , \infty-$\lambda$_{0} \cup i[-\sqrt{$\lambda$_{1} , \sqrt{ $\lambda$}]1) ,. and. m. has. m(z)=z+m_{1}z^{-1}+m_{2}z^{-2}+\cdots holds. Set. \left{bginary}{l \mathcl{Q}_$\ambd$_{0},\lambd$_{1}=\q;mathr{i}\mathr{n}\mathr{f}\mathr{s}\mathr{p}L_q>$\lambd$_{0}\mathr{}\mathr{n}\mathr{d}q\mathr{i}\mathr{s}\mathr{}\mathr{e}\mathr{f}\mathr{l}\mathr{e}\mathr{c}\mathr{}\mathr{i}\mathr{o}\mathr{n}\mathr{l}\mathr{e}\mathr{s}\mathr{s}\mathr{o}\mathr{n}($\lambd$_{1},\infty)}\ $Gam $=\{g; e^h}\matr{w}\mathr{i}\mathr{}\mathr{}\mathr{o}\mathr{d}\mathr{d}\mathr{p}\mathr{o}\mathr{l}\mathr{y}\mathr{n}\mathr{o}\mathr{}\mathr{i}\mathr{}\mathr{l}\mathr{s}\mathr{}\mathr{}\mathr{i}\mathr{s}\mathr{f}\mathr{y}\mathr{i}\mathr{n}\mathr{g}(z)=\overlin{h(\overlin{z})\ end{ary}\ight.. Let C, C' be simple closed curves surrounding the interval [-$\lambda$_{1}, -$\lambda$_{0}] counter‐ clockwise. The figure \mathrm{f}.1 indicates the situation.. \mathrm{f}.1.
(3) 57. For a function f denote by f_{e}, f_{0} the even part and odd part respectively, namely. f_{e}(z)=\displaystyle \frac{f(\sqrt{z})+f(-\sqrt{z})}{2}, f_{0}(z)=\frac{f(\sqrt{z})-f(-\sqrt{z})}{2\sqrt{z} . $\delta$. For whose $\delta$_{e}, $\delta$_{o} are holomorphic in a simply connected domain including C', set \tilde{m}(z)=m(z)- $\delta$(z) , and define. \left{bginary}l M_{g(z,$\lambd)=frc{\hatg}_o(z)\verlin{m})_($\labd)+ht{g}_e(z\ild{m})_0($\labd)}{ ma$-z}\ N_{g(,$lambd)=\frc{1}2$pi\nt_{C'}fracMg($\lmbda', $)}{\lambd-z}_{o($\lambd')^{-1}$\lambd' (N_{m}g)fz=\rac{1}2$pi\nt_{C}Ng(z,$\lambd)f( a$d\lmb. end{ary}\ight.. (4). for g\in $\Gamma$ and f \in L^{2}(C) , where \hat{g}(z) =g(z)^{-1} . Then, N_{m}(g) defines a trace class operator on L^{2}(C) . In [Ko2] we announced the following Theorem 1 For q\in Q_{$\lambda$_{0},$\lambda$_{1} and g\in $\Gamma$ define a tau‐function by. $\tau$_{m}(g)=\det(I+N_{m}(g)). .. Then, $\tau$_{m}(g)>0 is always valid, and. (K(g)q)(x)=-2\partial_{x}^{2}\log$\tau$_{m} (gex) with e_{x}(z)=e^{XZ} defines a smooth flow on \mathcal{Q}_{$\lambda$_{0},$\lambda$_{1} such that. \left\{\begin{ar ay}{l} (K(e^{tz})q)(x)=q(x+t) ,\ (K(e^{-4tz^{3} )q)(x) satisfies the Kd V equation. \end{ar ay}\right.. The class \mathcal{Q}_{$\lambda$_{0},$\lambda$_{1} contains multi‐solitons, algebro‐geometric solutions and. they are dense in \mathcal{Q}_{$\lambda$_{0},$\lambda$_{1} . Since $\tau$_{W}(e_{x}) is entire as a function of to be meromorphic on the entire complex plane C.. 2. x,. q(x) turns. Main result. Theorem 1 suggests the possibility to define $\tau$_{m} for general Weyl functions having no analyticity in a vicinity of \infty . We assume for simplicity \mathrm{i}\mathrm{n}\mathrm{f}\mathrm{s}\mathrm{p}L_{q}>-\infty and fix $\lambda$_{0} <\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{s}\mathrm{p}L_{q} . If q has no reflectionless property on any right half axis,. C, C' to -\infty . In order to have N_{m}(g) as a bounded e^{h} and should remain bounded along C, C' . Suppose $\Gamma$\ni g h(z)=h_{1}z^{n}+ lower order term with odd n , and for a>-$\lambda$_{0} let \{ $\omega$(x)\}_{x\leq a} be a continuous function satisfying. we have to extend the curves. operator. g_{e}, g_{0}. =. \left{\begin{ar y}{l $\omega$(x)>0\mathr{o}\mathr{n}(-\ifty,a)\ $\omega$(x)=0\mathr{o}\mathr{n}[a,\infty)\ $\omega$(x)=-^{$\alph$}\mathr{f}\mathr{o}\mathr{}x\leq-1 \end{ar y}\ight.. We assume the curve C is symmetric with respect to the real line, namely C=\overline{C} , and C is parametrized by $\omega$ by C. The shape of. C. on. c_{+}=\{x+i $\omega$(x); x\leq a\}.. is illustrated in f.2..
(4) 58. f.2 D. denotes the outside region of. C,. that is. D=\{z\in C; |{\rm Im} z| \geq $\omega$({\rm Re} z)\}. Since. h(\sqrt{z}) =h_{1}z^{n/2}+ lower order term, and for $\alpha$>-1. (x+i(-x)^{- $\alpha$})^{n/2}=i(-x)^{n/2}(1-\displaystyle \frac{n}{2}i(-x)^{- $\alpha$-1}+O(-x)^{-2( $\alpha$+1)}) holds as. x\rightarrow. -\infty,. e^{h(\pm\sqrt{z})} remain bounded as. $\alpha$-1\leq 0 , hence so do. {\rm Re} z. \rightarrow. -\infty. along. and g_{0}. Keeping this situation in mind, we define a metric. C. if n/2-. g_{e}. d_{ $\alpha,\ \beta$}(m_{1}, m_{2})=\displaystyle \sup_{z\in D}|z|^{ $\beta$}(|m_{1,e}(z)-m_{2,e}(z)|+|m_{1,0}(z)-m_{2,0}(z)|) for $\alpha$>-1, $\beta$>0 . We denote this whose Weyl functions m\pm satisfy. C. by C_{ $\alpha$} and let \mathcal{Q} be a set of all potentials. \left\{ begin{ar ay}{l m_{+}(z)=&\sqrt{-z}+\sum_{m=1}^{n}a_{m}(\sqrt{-z})^{-m}+o\ m_{-}(z)=&\sqrt{-z}+\sum_{\primen=1}^{n}b_{m}(\sqrt{-z})^{-m}+o \end{ar ay}\right\} {. for any n\geq satisfying. 1. along the curve C_{ $\alpha$} for any. a_{m}=b_{7n}. for odd. m. $\alpha$. > 0. (5). with real constants \{a_{m}, b_{m}\}. , and a_{ $\tau$ n}=-b_{m}. for even. m.. It should be noted that the property (5) holds on any sector \{ $\epsilon$<{\rm Im} z< $\pi$- $\epsilon$\} for any $\epsilon$>0 if q is smooth at x=0 . Clearly \mathcal{Q}_{$\lambda$_{0},$\lambda$_{1} \subset \mathcal{Q} is valid. The first key. lemma is. Lemma 1 (i) \mathcal{Q}_{$\lambda$_{\mathrm{O} ,$\lambda$_{1} is dense in \mathcal{Q} with metnc d_{ $\alpha,\ \beta$} for any $\alpha$, $\beta$>0. (ii) $\tau$_{m}(g) is conhnuous in m with respect to d_{ $\alpha,\ \beta$} for every sufficiently large $\beta$ if. g. is fixed.. Proof. Suppose. m\pm. are given by. m\displaystyle \pm(z)=i\sqrt{z}+\int_{-\infty}^{\infty}\frac{1}{ $\xi$-z}( $\sigma$\pm(d $\xi$)-\frac{\sqrt{ $\xi$}+}{ $\pi$}d $\xi$) and for. r> 1. ,. set. m_{\pm}^{r}(z)=i\displaystyle \sqrt{z}+\int_{r}^{\infty}\frac{ $\rho$( $\xi$)}{ $\xi$-z}d $\xi$+\int_{-r}^{r-1}\frac{1}{ $\xi$-z}( $\sigma$\pm(d $\xi$)-\frac{1}{ $\pi$}\sqrt{ $\xi$}+d $\xi$). ,. $\alpha$,.
(5) 59. where. $\rho$( \xi$)=\displaystyle\frac{\sqrt{$\xi$-r}{2$\pi$}\int_{-r}^{r-1}\frac{1}{$\xi-\xi$'}\frac{$\sigma$_{+}(d$\xi$')+$\sigma$_{-}(d$\xi$')-\frac{2}{$\pi$}\sqrt{$\xi$'}+d$\xi$'}{\sqrt{ -$\xi$'}. Then, the associated q_{r} is reflectionless on (r, \infty) . Under the condition on q one can show m^{r} \rightarrow m as r\rightarrow\infty . In the process of the proof we use a conformal map $\phi$ from \mathrm{c}_{+} onto D\cap C_{+}. \blacksquare. This lemma makes it possible to extend the definition of $\tau$_{m}(g) to that the associated q is an element of \mathcal{Q} , and we have Theorem 2 The extended $\tau$_{m}(g) sahsfies $\tau$_{m}(g). >0. such. m. and. (K(g)q)(x)=-2\partial_{x}^{2}\log$\tau$_{m} (gex) with e_{x}(z)=e^{XZ} defines a flow on \mathcal{Q} , namely K(g_{1}g_{2}) =K(g_{1})K(g_{2}) holds for any g_{1}, g_{2} \in $\Gamma$. If we choose g_{t}(z) =e^{tz} , then (K(g_{t})q)(x) =q(t+x) , and for g_{t}(z) =e^{-4tz^{3}} u(t, x)=(K(g_{t})q)(x) yields a solution to the Kd V equation. The definition of \mathcal{Q} is indirect, so the next task is to give simple sufficient conditions for q to be an element of \mathcal{Q}.. 3. Sufficient conditions. We have to find a rich family of potentials. q. satisfying (5). The first example is a. potential of the Schwartz space S(R) . In this case one can use the Jost solutions to estimate m\pm and without difficulty we have S(R) \subset \mathcal{Q} . The second example. is an almost periodic potential. We consider here a much wider class of ergodic potentials. To examine (5) we introduce the other two Herglotz functions. m_{1}(z)=-\displaystyle \frac{1}{m_{+}(z)+m_{-}(z)}, m_{2}(z)=\frac{m_{+}(z)m_{-}(z)}{m_{+}(z)+m_{-}(z)}. Observe that the condition (5) on m\pm \mathrm{i}\mathrm{s} equivalent a similar condition on. m_{1,2},. and that condition is achieved when $\xi$_{j}(z)=(\arg m_{j}(z))/ $\pi$\in [0 , 1 ] satisfy. \displaystyle\int_{0}^{\infty}$\lambda$^{n}|$\xi$_{j}($\lambda$)-\frac{1}{2}|d$\lambda$<\infty. (6). for any n\geq 1 and j=1 , 2. The function $\xi$_{1}( $\lambda$) was used by [GS1] and [GS2] to investigate the inverse spectral problem. To examine the condition (6) we need another quantity called reflection coefficient defined by. R(z)=\displaystyle \frac{m_{+}(z)+\overline{m_{-}(z)} {m_{+}(z)+m_{-}(z)}, which satisfies |R(z)|\leq 1 . This quantity was introduced by [Rybl] and studied by [Rem],.
(6) 60. Lemma 2 Suppose m\pm \in c_{+} and define m_{1,2} as above. Let $\xi$_{j} (\arg m_{j})/ $\pi$ and R=(m_{+}+\overline{m_{-}})/(m_{+}+m Then, the inequalities below are valid. =. |$\xi$_{1}-\displaystyle \frac{1}{2}|, $\xi$_{2}-\frac{1}{2}| \leq \frac{2}{ $\pi$}|R|. Therefore, (6) is reduced to. \displaystyle \int_{0}^{\infty}$\lambda$^{n}|R( $\lambda$)|d $\lambda$<\infty .. (7). For a general ergodic potential \{q_{ $\omega$}(x)\}_{ $\omega$\in $\Omega$} the non‐negative quantity $\gamma$(z) called Lyapounov exponent is crucial to investigate the spectrum of L_{q_{ $\omega$} . This. exponent is defined as. $\gamma$(z)=\displaystyle \lim_{x\rightar ow\infty}\frac{1}{x}\log\Vert U_{ $\omega$}(x, z where. U_{ $\omega$}(x, z). ,. is the 2\times 2 matrix solution to. \displaystyle \frac{d}{dx}U(x)= \left(\begin{ar ay}{l } 0 & \mathrm{l}\ z-q_{ $\omega$}(x) & 0 \end{ar ay}\right)U(x) , U(0)=I. Due to the ergodicity it is known that this limit exists a.s. $\omega$ . Floquet exponent w(z) which is an analog of the quantity in the case of periodic potentials is also defined by. w(z)=\mathrm{E}(m_{+, $\omega$}(z)) and it is known that. ,. $\gamma$(z)=-{\rm Re} w(z) . Set. $\chi$(z)=\displaystyle \frac{ $\gamma$(z)}{ \rm Im} z}-{\rm Im} w'(z). .. Then, [Kol] applied an identity. 4 $\chi$(z)=\displaystyle \mathrm{E}(|R(z)|^{2}(\frac{1}{ \rm Im} m_{+}(z)}+\frac{1}{ \rm Im} m_{-}(z)}). (8). to the study of the absolutely continuous spectrum of L_{q_{ $\omega$} . Schwarz inequality. together with (8) implies. \mathrm{E}(|R(z)|)\leq\sqrt{2 $\chi$(z){\rm Im} w(z)} .. (9). To apply (9) to the estimate (7) we have to use the conformal map $\phi$ again to shift the argument on the real axis to the one on the curve C_{ $\alpha$} . Consequently we have. Theorem 3 \mathcal{Q} contains the following potentials.. (i) the Schwartz space S(R) (ii) ergodic potentials \{q_{ $\omega$}(x)\}_{ $\omega$\in $\Omega$} satisfying. \displaystyle\int_{0}^{\infty}$\lambda$^{n}$\gam a$($\lambda$)d$\lambda$<\infty for any n\geq 1 . This condition is satisfied if \{q_{ $\omega$}(x)\}_{ $\omega$\in $\Omega$}\subset C_{b}^{\infty}(R) .. (iii) any smooth potential q which coincides with an element of S(R) on a half axis and coincides with an ergodic potential satisfying the condition in (ii) on. the other half axns..
(7) 61. Remark 1 If we are interested only in the. KdV. equation, we have only to. consider g_{t}(z) =e^{-4tz^{3}} , which relaxes the requirement that (6) should hold for all n\geq 1 to a requirement that (6) holds up to a certain fixed number N.. Almost periodic functions in the Bohr’s sense can be regarded as ergodic processes, hence Theorem 3 implies that the \mathrm{K}\mathrm{d}\mathrm{V} equation with smooth almost periodic functions can be solved globally. However, the almost periodicity of the solution remains open to be proved. Acknowledgement 1 Thi_{\mathcal{S}} research was partly supported by JSPS KAKENHI Grant Number 26400128.. References [BDGL]. I. Binder, D. Damanik, M. Goldstein, M. Lukic: Almost Penodicity in Time of Solutions of the KdV Equation, arXiv:1509.07373. [CKSTT] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao: Sharp global well‐posedness for KdV and modified KdV on R and T, Journal of. the AMS, 16 (2003), 705‐749. [DG]. D. Damanik‐M. Goldstein, On the existence and uniqueness of global solutions of the KdV equation with quasiperiodic initial data, J. Amer.. Math. Soc., 29 (2016), 825‐856 [Eg]. I. E. Egorova, The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense, Adv. Soviet. Math., 19 (1994), 181‐208. [GGKM] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura: A method for solving the Korteweg‐de Vries equation, Phys. Rev. Lett. 19 (1967) 1095 ‐ 1097.. [GS1] [GS2]. F. Gesztesy‐ B. Simon: The xi function, Acta Math., 176 (1996), 49‐71 $\Gamma$ .. Gesztesy‐ B. Simon: A new approach to inverse spectral theory,. II. General real potentials and the connection to the spectral measure,. Annals of Math., 152 (2000), 593‐643. [GR]. S. Grudsky‐A. Rybkin: Soliton Theory and Hankel Operators, SIAM J. Math. Anal., 47(2015), 2283‐2323.. [Kol]. S. Kotani: Ljapounov indices determine absolutely continuous spec‐ tra of stationary random Schrödinger operators, Stochastic Analysis. (Katata/Kyoto, 1982), 225‐ 247, North‐Holland Math. Library, 32, North‐Holland, Amsterdam, 1984.. [Ko2]. S. Kotani: Determinantal f_{07}mula of inverse spectral problem for Schrödinger operators and its application to KdV flow: Proceedings of V International Conference Analysis and Mathematical Physics 19‐23 June, 2017, Kharkiv, Ukraine.
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